Axiomatic characterizations of (G,O)-fuzzy rough approximation operators via overlap and grouping functions on a complete lattice

Recently, Jiang, H. B., and B. Q. Hu. [2022. “On (O,G)-Fuzzy Rough Sets Based on Overlap and Grouping Functions Over Complete Lattices.” International Journal of Approximate Reasoning 144: 18–50. doi:10.1016/j.ijar.2022.01.012] constructed a (Formula presented.) -fuzzy rough set model with the logical connectives–a grouping function (Formula presented.) and an overlap function (Formula presented.) on a complete lattice, which provided a new constructive approach to fuzzy rough set theory. The axiomatic approach is as important as the constructive approach in rough set theory. In this paper, we continue to study axiomatic characterizations of (Formula presented.) -fuzzy rough set. Traditionally, the associativity of the logical connectives plays a vital role in the axiomatic research of existing fuzzy rough set models. However, a grouping function (Formula presented.) and an overlap function (Formula presented.) lack the associativity. So we explore a novel axiomatic approach to (Formula presented.) -upper and (Formula presented.) -lower fuzzy rough approximation operators without associativity. Further, we provide single axioms to characterize (Formula presented.) -upper and (Formula presented.) -lower fuzzy rough approximation operators instead of sets of axioms. Finally, we use single axioms to characterize fuzzy rough approximation operators generated by various kinds of fuzzy relations including serial, reflexive, symmetric, (Formula presented.) -transitive, (Formula presented.) -transitive fuzzy relations as well as their compositions.